where the SU(3) flavor symmetry parameters \(f_i\equiv (f_i)^{L(R),s_p,s_k}_{VA,\lambda }\)  and \(\widetilde{f}_i\equiv (\widetilde{f}_i)^{L(R),s_p,s_k}_{VA,\lambda }\) . The \(f_i\)  terms in Eq. (REF ) and the later \(g_i\)  terms in Eq. (REF ) denote the short distance (SD) and the LD contributions via the single-quark transitions.
The \(\widetilde{f}_i\)  terms in Eq. (REF ) and the later \(\widetilde{g}_i\)  terms in Eq. (REF ) denote the \(W\) -exchange contributions of the two-quark and three-quark
transitions. \(T^{\prime }(\bar{3})=(1,0,0)\)  denotes the transition operators \((\bar{q}_2c)\)  with \(q_2=u\) , and \(H(\bar{6})^{lk}_j\)  (\(H(15)^{lk}_j\) ) related to the \((\bar{q}_lq^j)(\bar{q}_kc)\)  operator is antisymmetric (symmetric) in upper indices.
The nonvanishing \(H(\bar{6})^{lk}_j\)  and \(H(15)^{lk}_j\)  for \(c\rightarrow su\bar{d},du\bar{s},u\bar{d}d,u\bar{s}s\)  transitions can be found in Ref. [1]}.
Using \(l,k\)  antisymmetric in \(H(\bar{6})^{lk}_j\)  and \(l,k\)  symmetric in \(H(15)^{lk}_j\) , we have
\(\widetilde{f}_2=-\widetilde{f}_1, ~~~\widetilde{f}_5=\widetilde{f}_4, ~~~\widetilde{f}_6=0,\) 
